Geometrical optics | How light and sound interact with matter | Chem/phys

Reflection and refraction

Reflection is what happens when light bounces off a surface. In a mirror, light is almost completely reflected, and the angle of incidence equals the angle of reflection. When light hits the boundary between one medium and another, some of the light is reflected and the rest is transmitted into the second medium.

Refraction is the change in direction (bending) of light as it passes between media with different refractive indices ( $n$ ). The amount of bending is described by Snell’s law:

$n_{1} s in θ_{1}$ = $n_{2} s in θ_{2}$

Here:

$n_{1}$ and $n_{2}$ are the refractive indices of the two media.
$θ_{1}$ and $θ_{2}$ are measured from the normal, a line drawn perpendicular to the boundary surface.

Dispersion

Dispersion happens because a medium’s refractive index depends on wavelength. That means different colors bend by different amounts when they enter the medium. In a prism, blue light (shorter wavelength) refracts more than red light (longer wavelength), so white light spreads out into a spectrum.

Conditions for total internal reflection

Total internal reflection occurs when:

light travels from a medium with a higher refractive index to one with a lower refractive index, and
the angle of incidence is greater than the critical angle.

The critical angle $θ_{c}$ is found from:

n₁ sinθc = n₂

where $n_{1}$ is the refractive index of the denser (higher- $n$ ) medium and $n_{2}$ is the refractive index of the less dense (lower- $n$ ) medium. When the incident angle exceeds $θ_{c}$ , no refracted ray enters the second medium; instead, the light is completely reflected back into the denser medium.

Spherical mirrors

Spherical mirrors are reflective surfaces shaped like part of a sphere. They come in two forms: concave mirrors and convex mirrors:

A concave mirror has an inward curvature and focuses parallel light rays to a focal point. It’s a converging mirror with a positive focal length.
A convex mirror has an outward curvature and makes light rays diverge. It forms virtual images behind the mirror and has a negative focal length. The radius of curvature is the distance from the mirror to the center of the sphere it came from, and the focal length is half of this value ( $f = R /2$ ).

The imaging properties are described by the mirror equation:

$1/ p + 1/ q = 1/ f$

where object distance ( $p$ ) is measured from the mirror to the object and image distance ( $q$ ) is measured from the mirror to the image. By convention:

$p$ is positive.
$q$ is positive for a real image (formed in front of the mirror) and negative for a virtual image (formed behind the mirror).

Magnification ( $M$ ) is defined as $M = h^{'} / h = - q / p$ , where $h^{'}$ is the image height and $h$ is the object height. Real images formed by concave mirrors are inverted, while virtual images are upright. Convex mirrors are diverging, so they always produce virtual images.

Concave mirror ray diagrams illustrating image formation

Thin lenses

Thin lenses form images by transmitting and bending light (refraction), rather than reflecting it. There are two primary types: convex lenses (converging) and concave lenses (diverging).

A convex lens is similar to a concave mirror in that it converges light rays. Real images form on the opposite side of the lens from the object because the rays actually converge after passing through the lens. Virtual images appear on the same side as the object because the rays do not actually converge.

Concave lenses make light rays diverge, so they produce only virtual images. These images are upright and form on the same side of the lens as the object.

The focal length of a lens tells you where parallel rays converge (or appear to diverge). For a converging lens, the focal length is positive; for a diverging lens, it is negative.

The relationship between the object distance ( $P$ ), the image distance ( $q$ ), and the focal length ( $f$ ) is:

$1/ p + 1/ q = 1/ f$

Using these sign conventions:

$P$ is always positive.
$q$ is positive for real images and negative for virtual images.

The strength (or power) of a lens is measured in diopters ( $P$ ) and is defined by $P = 1/ f$ , with $f$ in meters. Lenses can also show aberrations, such as spherical aberration (not all rays focus at one point) and chromatic aberration (different wavelengths refract by different amounts, producing color fringing).

Combination of lenses

In an optical system, a real image formed by one lens can act as the object for a second lens. For multiple lenses, the overall magnification is the product of the individual magnifications from each lens.

Lens aberration

Lens aberration refers to imperfections in an optical system that produce blurred, distorted, or color-fringed images:

Spherical aberration arises when rays passing through different lens regions converge at multiple points, reducing sharpness.
Chromatic aberration occurs because the lens cannot focus all wavelengths together, creating color fringing.
Astigmatism reflects varying focal lengths along different axes, causing uneven sharpness. Distortion means magnification changes with distance from the lens center, bending straight lines. Field curvature implies the lens projects onto a non-flat plane, blurring image edges.
Coma leads off-axis points to appear comet-shaped, reducing clarity near the field’s periphery.

Optical instruments

The eye works as an optical instrument by using a lens to focus a real image onto the retina.

Glasses correct vision by using either a diverging (concave) lens for near-sightedness or a converging (convex) lens for far-sightedness.

A magnifying glass is a converging lens. When the object distance ( $p$ ) is less than the focal length ( $f$ ), it forms a virtual, erect, and enlarged image.

Reflection and refraction

Reflection: angle of incidence = angle of reflection
Refraction: bending of light at boundary between media with different refractive indices
Snell’s law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$

Dispersion

Refractive index varies with wavelength
Blue light bends more than red in a prism, creating a spectrum

Conditions for total internal reflection

Occurs when light moves from higher to lower refractive index
Angle of incidence > critical angle ( $n_{1} sin θ_{c} = n_{2}$ )
All light reflected back into denser medium

Spherical mirrors

Concave: converging, positive focal length, real/inverted or virtual/upright images
Convex: diverging, negative focal length, always virtual/upright images
Mirror equation: $1/ p + 1/ q = 1/ f$
Magnification: $M = h^{'} / h = - q / p$

Thin lenses

Convex (converging): positive focal length, real or virtual images
Concave (diverging): negative focal length, only virtual images
Lens equation: $1/ p + 1/ q = 1/ f$
Power: $P = 1/ f$ (f in meters), measured in diopters
Aberrations: spherical (blur), chromatic (color fringing)

Combination of lenses

Image from first lens can be object for second lens
Total magnification = product of individual magnifications

Lens aberration

Spherical aberration: rays focus at different points, reduced sharpness
Chromatic aberration: different wavelengths focus differently, color fringing
Astigmatism: uneven sharpness along axes
Distortion: magnification varies with distance from center
Field curvature: image not flat, blurred edges
Coma: off-axis points appear comet-shaped

Optical instruments

Eye: lens forms real image on retina
Glasses: concave lens for near-sightedness, convex lens for far-sightedness
Magnifying glass: converging lens, forms virtual, erect, enlarged image when $p < f$

Reflection and refraction

Refraction is the change in direction (bending) of light as it passes between media with different refractive indices ( $n$ ). The amount of bending is described by Snell’s law:

$n_{1} s in θ_{1}$ = $n_{2} s in θ_{2}$

Here:

$n_{1}$ and $n_{2}$ are the refractive indices of the two media.
$θ_{1}$ and $θ_{2}$ are measured from the normal, a line drawn perpendicular to the boundary surface.

Dispersion

Conditions for total internal reflection

Total internal reflection occurs when:

light travels from a medium with a higher refractive index to one with a lower refractive index, and
the angle of incidence is greater than the critical angle.

The critical angle $θ_{c}$ is found from:

n₁ sinθc = n₂

Spherical mirrors

Spherical mirrors are reflective surfaces shaped like part of a sphere. They come in two forms: concave mirrors and convex mirrors:

A concave mirror has an inward curvature and focuses parallel light rays to a focal point. It’s a converging mirror with a positive focal length.
A convex mirror has an outward curvature and makes light rays diverge. It forms virtual images behind the mirror and has a negative focal length. The radius of curvature is the distance from the mirror to the center of the sphere it came from, and the focal length is half of this value ( $f = R /2$ ).

The imaging properties are described by the mirror equation:

$1/ p + 1/ q = 1/ f$

where object distance ( $p$ ) is measured from the mirror to the object and image distance ( $q$ ) is measured from the mirror to the image. By convention:

$p$ is positive.
$q$ is positive for a real image (formed in front of the mirror) and negative for a virtual image (formed behind the mirror).

Thin lenses

Thin lenses form images by transmitting and bending light (refraction), rather than reflecting it. There are two primary types: convex lenses (converging) and concave lenses (diverging).

A convex lens is similar to a concave mirror in that it converges light rays. Real images form on the opposite side of the lens from the object because the rays actually converge after passing through the lens. Virtual images appear on the same side as the object because the rays do not actually converge.

Concave lenses make light rays diverge, so they produce only virtual images. These images are upright and form on the same side of the lens as the object.

The focal length of a lens tells you where parallel rays converge (or appear to diverge). For a converging lens, the focal length is positive; for a diverging lens, it is negative.

The relationship between the object distance ( $P$ ), the image distance ( $q$ ), and the focal length ( $f$ ) is:

$1/ p + 1/ q = 1/ f$

Using these sign conventions:

$P$ is always positive.
$q$ is positive for real images and negative for virtual images.

Combination of lenses

Lens aberration

Lens aberration refers to imperfections in an optical system that produce blurred, distorted, or color-fringed images:

Spherical aberration arises when rays passing through different lens regions converge at multiple points, reducing sharpness.
Chromatic aberration occurs because the lens cannot focus all wavelengths together, creating color fringing.
Astigmatism reflects varying focal lengths along different axes, causing uneven sharpness. Distortion means magnification changes with distance from the lens center, bending straight lines. Field curvature implies the lens projects onto a non-flat plane, blurring image edges.
Coma leads off-axis points to appear comet-shaped, reducing clarity near the field’s periphery.

Optical instruments

The eye works as an optical instrument by using a lens to focus a real image onto the retina.

Glasses correct vision by using either a diverging (concave) lens for near-sightedness or a converging (convex) lens for far-sightedness.

A magnifying glass is a converging lens. When the object distance ( $p$ ) is less than the focal length ( $f$ ), it forms a virtual, erect, and enlarged image.

Reflection and refraction

Reflection: angle of incidence = angle of reflection
Refraction: bending of light at boundary between media with different refractive indices
Snell’s law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$

Dispersion

Refractive index varies with wavelength
Blue light bends more than red in a prism, creating a spectrum

Conditions for total internal reflection

Occurs when light moves from higher to lower refractive index
Angle of incidence > critical angle ( $n_{1} sin θ_{c} = n_{2}$ )
All light reflected back into denser medium

Spherical mirrors

Concave: converging, positive focal length, real/inverted or virtual/upright images
Convex: diverging, negative focal length, always virtual/upright images
Mirror equation: $1/ p + 1/ q = 1/ f$
Magnification: $M = h^{'} / h = - q / p$

Thin lenses

Convex (converging): positive focal length, real or virtual images
Concave (diverging): negative focal length, only virtual images
Lens equation: $1/ p + 1/ q = 1/ f$
Power: $P = 1/ f$ (f in meters), measured in diopters
Aberrations: spherical (blur), chromatic (color fringing)

Combination of lenses

Image from first lens can be object for second lens
Total magnification = product of individual magnifications

Lens aberration

Spherical aberration: rays focus at different points, reduced sharpness
Chromatic aberration: different wavelengths focus differently, color fringing
Astigmatism: uneven sharpness along axes
Distortion: magnification varies with distance from center
Field curvature: image not flat, blurred edges
Coma: off-axis points appear comet-shaped

Optical instruments

Eye: lens forms real image on retina
Glasses: concave lens for near-sightedness, convex lens for far-sightedness
Magnifying glass: converging lens, forms virtual, erect, enlarged image when $p < f$

Reflection and refraction

Refraction is the change in direction (bending) of light as it passes between media with different refractive indices ( $n$ ). The amount of bending is described by Snell’s law:

$n_{1} s in θ_{1}$ = $n_{2} s in θ_{2}$

Here:

$n_{1}$ and $n_{2}$ are the refractive indices of the two media.
$θ_{1}$ and $θ_{2}$ are measured from the normal, a line drawn perpendicular to the boundary surface.

Dispersion

Conditions for total internal reflection

Total internal reflection occurs when:

light travels from a medium with a higher refractive index to one with a lower refractive index, and
the angle of incidence is greater than the critical angle.

The critical angle $θ_{c}$ is found from:

n₁ sinθc = n₂

Spherical mirrors

Spherical mirrors are reflective surfaces shaped like part of a sphere. They come in two forms: concave mirrors and convex mirrors:

A concave mirror has an inward curvature and focuses parallel light rays to a focal point. It’s a converging mirror with a positive focal length.
A convex mirror has an outward curvature and makes light rays diverge. It forms virtual images behind the mirror and has a negative focal length. The radius of curvature is the distance from the mirror to the center of the sphere it came from, and the focal length is half of this value ( $f = R /2$ ).

The imaging properties are described by the mirror equation:

$1/ p + 1/ q = 1/ f$

where object distance ( $p$ ) is measured from the mirror to the object and image distance ( $q$ ) is measured from the mirror to the image. By convention:

$p$ is positive.
$q$ is positive for a real image (formed in front of the mirror) and negative for a virtual image (formed behind the mirror).

Thin lenses

Thin lenses form images by transmitting and bending light (refraction), rather than reflecting it. There are two primary types: convex lenses (converging) and concave lenses (diverging).

A convex lens is similar to a concave mirror in that it converges light rays. Real images form on the opposite side of the lens from the object because the rays actually converge after passing through the lens. Virtual images appear on the same side as the object because the rays do not actually converge.

Concave lenses make light rays diverge, so they produce only virtual images. These images are upright and form on the same side of the lens as the object.

The focal length of a lens tells you where parallel rays converge (or appear to diverge). For a converging lens, the focal length is positive; for a diverging lens, it is negative.

The relationship between the object distance ( $P$ ), the image distance ( $q$ ), and the focal length ( $f$ ) is:

$1/ p + 1/ q = 1/ f$

Using these sign conventions:

$P$ is always positive.
$q$ is positive for real images and negative for virtual images.

Combination of lenses

Lens aberration

Lens aberration refers to imperfections in an optical system that produce blurred, distorted, or color-fringed images:

Spherical aberration arises when rays passing through different lens regions converge at multiple points, reducing sharpness.
Chromatic aberration occurs because the lens cannot focus all wavelengths together, creating color fringing.
Astigmatism reflects varying focal lengths along different axes, causing uneven sharpness. Distortion means magnification changes with distance from the lens center, bending straight lines. Field curvature implies the lens projects onto a non-flat plane, blurring image edges.
Coma leads off-axis points to appear comet-shaped, reducing clarity near the field’s periphery.

Optical instruments

The eye works as an optical instrument by using a lens to focus a real image onto the retina.

Glasses correct vision by using either a diverging (concave) lens for near-sightedness or a converging (convex) lens for far-sightedness.

A magnifying glass is a converging lens. When the object distance ( $p$ ) is less than the focal length ( $f$ ), it forms a virtual, erect, and enlarged image.

Reflection and refraction

Reflection: angle of incidence = angle of reflection
Refraction: bending of light at boundary between media with different refractive indices
Snell’s law: $n_{1} sin θ_{1} = n_{2} sin θ_{2}$

Dispersion

Refractive index varies with wavelength
Blue light bends more than red in a prism, creating a spectrum

Conditions for total internal reflection

Occurs when light moves from higher to lower refractive index
Angle of incidence > critical angle ( $n_{1} sin θ_{c} = n_{2}$ )
All light reflected back into denser medium

Spherical mirrors

Concave: converging, positive focal length, real/inverted or virtual/upright images
Convex: diverging, negative focal length, always virtual/upright images
Mirror equation: $1/ p + 1/ q = 1/ f$
Magnification: $M = h^{'} / h = - q / p$

Thin lenses

Convex (converging): positive focal length, real or virtual images
Concave (diverging): negative focal length, only virtual images
Lens equation: $1/ p + 1/ q = 1/ f$
Power: $P = 1/ f$ (f in meters), measured in diopters
Aberrations: spherical (blur), chromatic (color fringing)

Combination of lenses

Image from first lens can be object for second lens
Total magnification = product of individual magnifications

Lens aberration

Spherical aberration: rays focus at different points, reduced sharpness
Chromatic aberration: different wavelengths focus differently, color fringing
Astigmatism: uneven sharpness along axes
Distortion: magnification varies with distance from center
Field curvature: image not flat, blurred edges
Coma: off-axis points appear comet-shaped

Optical instruments

Eye: lens forms real image on retina
Glasses: concave lens for near-sightedness, convex lens for far-sightedness
Magnifying glass: converging lens, forms virtual, erect, enlarged image when $p < f$